6 problems

Let $k$ be a positive integer, let $d$ be the rank parameter for the tadpole Dynkin diagram, and let $T_d$ denote its Cartan matrix. Consider the level $k$ standard…

Charge-refined q-series conjecture. The following identity holds:

Let $W_{k,i,j}^{0alpha_1,0alpha_1+0alpha_2}=U(0\overline{\mathfrak{n}}[\hat{\nu}])\cdot v_{k,i,j}$, where … and $0\leq i,j\leq k$ with $i+j\leq k$. For $j=0$, these are the spaces…

Let $W_{k,i}$ be the virtual subspaces associated with level $k$ standard modules for $A_2^{(2)}$, and let their characters be denoted by $0chi_{W_{k,i}}'(x,q)$. Recursion conjectu…

Let $\mathfrak{g}=\mathfrak{sl}(n+1)$, let $\Lambda_i$ be the fundamental weights, and let $\bar{U(\mathfrak{n})}$, $\widetilde{U(\bar{\mathfrak{n}})\bar{\mathfrak{n}}_{+}}$,…

Let $\widehat{\mathfrak{sl}(n+1)}$ be the affine Lie algebra, let $L(\Lambda)$ be a standard module of positive integral level $k$ with highest-weight vector $v_\Lambda$, and let…